scholarly journals Cyclotomic quiver Hecke algebras and Hecke algebra of $G(r,p,n)$

2018 ◽  
Vol 371 (6) ◽  
pp. 3877-3916 ◽  
Author(s):  
Salim Rostam
Keyword(s):  
Hecke Algebras ◽  
Hecke Algebra

scholarly journals A quantum probabilistic approach to Hecke algebras for 𝔭-adic PGL2

2018 ◽  
Vol 21 (03) ◽  
pp. 1850015
Author(s):  
Takehiro Hasegawa ◽  
Hayato Saigo ◽  
Seiken Saito ◽  
Shingo Sugiyama
Keyword(s):  
Inversion Formula ◽  
Probabilistic Approach ◽  
Hecke Algebras ◽  
Hecke Algebra ◽  
Quantum Probability ◽  
Probabilistic Interpretation ◽  
Fourier Inversion ◽  
Fourier Inversion Formula ◽  
The Subject

The subject of the present paper is an application of quantum probability to [Formula: see text]-adic objects. We give a quantum-probabilistic interpretation of the spherical Hecke algebra for [Formula: see text], where [Formula: see text] is a [Formula: see text]-adic field. As a byproduct, we obtain a new proof of the Fourier inversion formula for [Formula: see text].

scholarly journals Murnaghan-Nakayama Rules for Characters of Iwahori-Hecke Algebras of the Complex Reflection Groups G(r, p, n)

1998 ◽  
Vol 50 (1) ◽  
pp. 167-192 ◽  
Author(s):  
Tom Halverson ◽  
Arun Ram
Keyword(s):  
Symmetric Group ◽  
Infinite Series ◽  
Hecke Algebras ◽  
Hecke Algebra ◽  
Diagonal Matrix ◽  
Weyl Groups ◽  
Reflection Groups ◽  
Irreducible Characters ◽  
Complex Reflection ◽  
Complex Reflection Groups

AbstractIwahori-Hecke algebras for the infinite series of complex reflection groups G(r, p, n) were constructed recently in the work of Ariki and Koike [AK], Broué andMalle [BM], and Ariki [Ari]. In this paper we give Murnaghan-Nakayama type formulas for computing the irreducible characters of these algebras. Our method is a generalization of that in our earlier paper [HR] in whichwe derivedMurnaghan-Nakayama rules for the characters of the Iwahori-Hecke algebras of the classical Weyl groups. In both papers we have been motivated by C. Greene [Gre], who gave a new derivation of the Murnaghan-Nakayama formula for irreducible symmetric group characters by summing diagonal matrix entries in Young's seminormal representations. We use the analogous representations of the Iwahori-Hecke algebra of G(r, p, n) given by Ariki and Koike [AK] and Ariki [Ari].

Integral Hecke Algebras for Finite Generalized Polygons

2011 ◽  
Vol 18 (02) ◽  
pp. 259-272
Author(s):  
İlhan Hacıoglu
Keyword(s):  
Normal Form ◽  
Incidence Matrix ◽  
Hecke Algebras ◽  
Hecke Algebra ◽  
Generalized Polygons ◽  
Completely Reducible ◽  
Standard Module ◽  
Rank 2

Suppose that (P,B,F) is a triple consisting of the points, blocks and flags of a generalized m-gon, and H(F) the associated rank-2 Iwahori–Hecke algebra. H(F) acts naturally on the integral standard module ZF based on F. This work gives arithmetic conditions on a subring R, where R contains the integers and is contained in the rationals, that insure the associated R-ary Iwahori–Hecke algebra to be completely reducible on RF. The constituent multiplicities are related to the R-normal form of the incidence matrix of (P,B,F).

scholarly journals HECKE ALGEBRAS FOR INNER FORMS OF -ADIC SPECIAL LINEAR GROUPS

2015 ◽  
Vol 16 (2) ◽  
pp. 351-419 ◽  
Author(s):  
Anne-Marie Aubert ◽  
Paul Baum ◽  
Roger Plymen ◽  
Maarten Solleveld
Keyword(s):  
Finite Group ◽  
Local Field ◽  
Hecke Algebras ◽  
Hecke Algebra ◽  
Linear Groups ◽  
Special Linear Groups ◽  
Affine Hecke Algebra ◽  
Morita Equivalent ◽  
Affine Hecke Algebras ◽  
A Finite Group

Let$F$be a non-Archimedean local field, and let$G^{\sharp }$be the group of$F$-rational points of an inner form of$\text{SL}_{n}$. We study Hecke algebras for all Bernstein components of$G^{\sharp }$, via restriction from an inner form$G$of$\text{GL}_{n}(F)$.For any packet of L-indistinguishable Bernstein components, we exhibit an explicit algebra whose module category is equivalent to the associated category of complex smooth$G^{\sharp }$-representations. This algebra comes from an idempotent in the full Hecke algebra of$G^{\sharp }$, and the idempotent is derived from a type for$G$. We show that the Hecke algebras for Bernstein components of$G^{\sharp }$are similar to affine Hecke algebras of type$A$, yet in many cases are not Morita equivalent to any crossed product of an affine Hecke algebra with a finite group.

scholarly journals R-POLYNOMIALS OF FINITE MONOIDS OF LIE TYPE

2010 ◽  
Vol 20 (06) ◽  
pp. 793-805 ◽  
Author(s):  
KÜRŞAT AKER ◽  
MAHIR BILEN CAN ◽  
MÜGE TAŞKIN
Keyword(s):  
Weyl Group ◽  
Hecke Algebras ◽  
Hecke Algebra ◽  
Möbius Function ◽  
Necessary Condition ◽  
Finite Monoid ◽  
Finite Monoids ◽  
Mobius Function ◽  
Renner Monoid

This paper studies the combinatorics of the orbit Hecke algebras associated with W × W orbits in the Renner monoid of a finite monoid of Lie type, M, where W is the Weyl group associated with M. It is shown by Putcha in [12] that the Kazhdan–Lusztig involution [6] can be extended to the orbit Hecke algebra which enables one to define the R-polynomials of the intervals contained in a given orbit. Using the R-polynomials, we calculate the Möbius function of the Bruhat–Chevalley ordering on the orbits. Furthermore, we provide a necessary condition for an interval contained in a given orbit to be isomorphic to an interval in some Weyl group.

scholarly journals Finiteness of cohomology groups of stacks of shtukas as modules over Hecke algebras, and applications

2020 ◽  
Vol Volume 4 ◽  
Author(s):  
Cong Xue
Keyword(s):  
Finite Type ◽  
Compact Support ◽  
Automorphic Forms ◽  
Hecke Algebras ◽  
Hecke Algebra ◽  
Constant Term ◽  
Cohomology Groups ◽  
Quotient Spaces

In this paper we prove that the cohomology groups with compact support of stacks of shtukas are modules of finite type over a Hecke algebra. As an application, we extend the construction of excursion operators, defined by V. Lafforgue on the space of cuspidal automorphic forms, to the space of automorphic forms with compact support. This gives the Langlands parametrization for some quotient spaces of the latter, which is compatible with the constant term morphism. Comment: published version

scholarly journals Products of Geck-Rouquier conjugacy classes and the Hecke algebra of composed permutations

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Pierre-Loïc Méliot
Keyword(s):  
Hecke Algebras ◽  
Hecke Algebra ◽  
Projective Limit ◽  
Conjugacy Classes ◽  
International Audience ◽  
Structure Constants

International audience We show the $q$-analog of a well-known result of Farahat and Higman: in the center of the Iwahori-Hecke algebra $\mathscr{H}_{n,q}$, if $(a_{\lambda \mu}^ν (n,q))_ν$ is the set of structure constants involved in the product of two Geck-Rouquier conjugacy classes $\Gamma_{\lambda, n}$ and $\Gamma_{\mu,n}$, then each coefficient $a_{\lambda \mu}^ν (n,q)$ depend on $n$ and $q$ in a polynomial way. Our proof relies on the construction of a projective limit of the Hecke algebras; this projective limit is inspired by the Ivanov-Kerov algebra of partial permutations. Nous démontrons le $q$-analogue d'un résultat bien connu de Farahat et Higman : dans le centre de l'algèbre d'Iwahori-Hecke $\mathscr{H}_{n,q}$, si $(a_{\lambda \mu}^ν (n,q))_ν$ est l'ensemble des constantes de structure mises en jeu dans le produit de deux classes de conjugaison de Geck-Rouquier $\Gamma_{\lambda, n}$ et $\Gamma_{\mu,n}$, alors chaque coefficient $a_{\lambda \mu}^ν (n,q)$ dépend de façon polynomiale de $n$ et de $q$. Notre preuve repose sur la construction d'une limite projective des algèbres d'Hecke ; cette limite projective est inspirée de l'algèbre d'Ivanov-Kerov des permutations partielles.

scholarly journals Characters of Renner monoids and their Hecke algebras

2020 ◽  
Vol 30 (07) ◽  
pp. 1505-1535
Author(s):  
Andrew Hardt ◽  
Jared Marx-Kuo ◽  
Vaughan McDonald ◽  
John M. O’Brien ◽  
Alexander Vetter
Keyword(s):  
Hecke Algebras ◽  
Hecke Algebra ◽  
Explicit Description ◽  
General Algorithm ◽  
Character Table ◽  
Cartan Type ◽  
Type Formula ◽  
Rook Monoid ◽  
Combinatorial Formulas ◽  
Renner Monoid

This paper gives a general algorithm for computing the character table of any Renner monoid Hecke algebra, by adapting and generalizing techniques of Solomon used to study the rook monoid. The character table of the Hecke algebra of the rook monoid (i.e. the Cartan type [Formula: see text] Renner monoid) was computed earlier by Dieng et al. [2], using different methods. Our approach uses analogues of so-called A- and B-matrices of Solomon. In addition to the algorithm, we give explicit combinatorial formulas for the A- and B-matrices in Cartan type [Formula: see text] and use them to obtain an explicit description of the character table for the type [Formula: see text] Renner monoid Hecke algebra.

scholarly journals ON REPRESENTATIONS OF CYCLOTOMIC HECKE ALGEBRAS

2011 ◽  
Vol 26 (11) ◽  
pp. 795-803 ◽  
Author(s):  
O. V. OGIEVETSKY ◽  
L. POULAIN D'ANDECY
Keyword(s):  
Representation Theory ◽  
Hecke Algebras ◽  
Hecke Algebra ◽  
Cyclotomic Hecke Algebra ◽  
Cyclotomic Hecke Algebras

An approach, based on Jucys–Murphy elements, to the representation theory of cyclotomic Hecke algebras is developed. The maximality (in the cyclotomic Hecke algebra) of the set of the Jucys–Murphy elements is established. A basis of the cyclotomic Hecke algebra is suggested; this basis is used to establish the flatness of the deformation without using the representation theory.

scholarly journals Identifying the Invariants for Classical Knots and Links from the Yokonuma–Hecke Algebras

2018 ◽  
Vol 2020 (1) ◽  
pp. 214-286 ◽  
Author(s):  
Maria Chlouveraki ◽  
Jesús Juyumaya ◽  
Konstantinos Karvounis ◽  
Sofia Lambropoulou
Keyword(s):  
Hecke Algebras ◽  
Hecke Algebra ◽  
Type A ◽  
Closed Formula ◽  
Link Invariant ◽  
Polynomial Invariants ◽  
Oriented Link ◽  
Linking Numbers ◽  
Markov Trace ◽  
Knots And Links

Abstract We announce the existence of a family of new 2-variable polynomial invariants for oriented classical links defined via a Markov trace on the Yokonuma–Hecke algebra of type A. Yokonuma–Hecke algebras are generalizations of Iwahori–Hecke algebras, and this family contains the HOMFLYPT polynomial, the famous 2-variable invariant for classical links arising from the Iwahori–Hecke algebra of type A. We show that these invariants are topologically equivalent to the HOMFLYPT polynomial on knots, but not on links, by providing pairs of HOMFLYPT-equivalent links that are distinguished by our invariants. In order to do this, we prove that our invariants can be defined diagrammatically via a special skein relation involving only crossings between different components. We further generalize this family of invariants to a new 3-variable skein link invariant that is stronger than the HOMFLYPT polynomial. Finally, we present a closed formula for this invariant, by W. B. R. Lickorish, that uses HOMFLYPT polynomials of sublinks and linking numbers of a given oriented link.

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